# How can a piece of A4 paper be folded in exactly three equal parts?

This is something that always annoys me when putting an A4 letter in a oblong envelope: one has to estimate where to put the creases when folding the letter. I normally start from the bottom and on eye estimate where to fold. Then I turn the letter over and fold bottom to top. Most of the time ending up with three different areas. There must be a way to do this exactly, without using any tools (ruler, etc.).

-
In case you print the letter: A more appropriate way is to print fold marks. In LaTeX, you get them when you use the scrlttr2 environment as shown here. –  moose Apr 2 at 14:40
This isn't a geometric solution, but as a practical way to fold approximate thirds: loosely fold the paper (so that it doesn't make the permanent creases/fold marks) in thirds, and then tightly fold it once you're satisfied with the alignment. –  Tim S. Apr 2 at 17:52
You normally don't want exactly equal parts. The upper part should be a bit longer than the other two, so you don't cut the inside paper when opening the letter. –  Paŭlo Ebermann Apr 2 at 20:52
I know this probably doesn't answer your question, but I usually just use the edge of the envelope as a guide. –  Erik Miehling Apr 3 at 19:18
A mathematician, a computer scientist, and an engineer were once faced with this very problem. The computer scientist began to work on a recursive algorithm that would allocate more paper to each segment until they met in the middle. The engineer just made a guess and sent the letter. The mathematician is still thinking about it. –  Ollie Ford Apr 6 at 17:04

Fold twice to obtain quarter markings at the paper bottom. Fold along the line through the top corner and the third of these marks. The vertical lines through the first two marks intersect this inclined line at thirds, which allows the final foldings.

(Photo by Ross Millikan below - if the image helped you, you can up-vote his too...)

-
Perfect! I'll use this method for all business correspondence from now on =) –  Jens Apr 2 at 14:02
And of course this has the advantage of utterly befuddling the recipient. –  episanty Apr 2 at 21:12
Keep a "template" like this handy so that you don't send the worlds most creased correspondence. –  Daniel E. Apr 3 at 0:04
1) With this method one can fold a letter an arbitrary number of times (if one does not reach physical limits ). So if one wants to fold it in 5 equal parts one first folds it in $2^3=8$ equal parts and the folds along the line through the top corner and the 5th mark. 2) One must use the second dimensions. When folding only parallel to one side one cannot fold into 3 equal parts. It is not possible to partition an interval $[0,1]$ in three equal parts by bisecting . The points that you can construct this way are $0,1$ and points of the form $\frac{n}{2^k}$. –  miracle173 Apr 3 at 5:43
My results are significantly better if I fold three times to get the quarter markings (fold in half, unfold, fold each half in half). Following the letter(!) of your method as currently written I am off by almost half an inch every time. The diagonal fold is fairly difficult to get right as well, since there is nothing with which to line up the end of the paper as you fold, just a page corner on one side and crease/border intersection on the other. I love the elegance and generalizability, but 5 or 6 folds make this about as reliable as the default "flatten the tube" method in practice. –  Tyler James Young Apr 3 at 19:22

Here is a picture to go with Hagen von Eitzen's answer. The horizontal lines are the result of the first two folds. The diagonal line is the third fold. The heavy lines are the points at thirds for folding into the envelope.

-
@Nate: here it is –  Ross Millikan Apr 2 at 15:36
Ross thanks! Clear now! –  Nicky Hekster Apr 2 at 17:10
Shouldn't this be an edit to the previous answer, not a separate answer by itself? –  Nit Apr 2 at 19:38
@RossMillikan If you edit someone's post their signature isn't overwritten, the original poster is still shown next to the edit information. –  Nit Apr 2 at 20:51
"over [someone]'s signature" doesn't mean "overwriting [someone]'s signature" –  Steven Taschuk Apr 3 at 15:30

This is both practical (no extra creases) and precise (no guessing or estimating).

1. Roll the paper into a 3-ply tube, with the ends aligned:

2. Pinch the paper (crease the edge) where I've drawn the red line

3. Unroll
4. Use the pinch mark to show where the folds should be
-
This is also nice in that it doesn't leave the paper with extraneous creases. –  Nate Eldredge Apr 3 at 19:22
No extra creases. Exactly the right place. Quick to execute. Works for any size paper. This is clearly the best answer. –  Sam Dickson Apr 4 at 13:00
But the "outermost" third will be slightly larger than the middle one (and that one than the innermost one), by at least $2\pi$ times the thickness of the paper. (More if rolling up actually resembles the illustration.) –  Marc van Leeuwen Apr 4 at 13:01
@MarcvanLeeuwen True, but for very important correspondence (that which must be folded exactly into thirds) one would generally use an ideal sheet of paper with 0 thickness, so that wouldn't be a problem. –  iamnotmaynard Apr 4 at 15:16
@MarcVanLeeuwen But as anyone who's experienced the pain of designing a tri-fold brochure will know, this is exactly what you want! (Well for those folded like a 6, not a Z.) The third that you fold first should be the smallest (by a mm or so) so that it's not squished and damaged inside the second fold. And it allows the final edge to align nicely with the first fold and not fall short. Love this solution! –  Simon Apr 4 at 23:30

You only need to fold once in half and fold two diagonals. The second line can be found from folding the first third over again.

-
Very nice and minimalist. –  Vincent Apr 7 at 11:43
1. Roll in to a cylinder until both edges are opposite to each other.
2. Fold the points where the edges touches the paper. (Squish the cylinder from left and right)

Approximation method:

1. Assume a 120 degree angle and fold as shown below.
2. For accuracy, match edge-side to any of the other two sides.
-
Estimating when the two ends are diametrically opposite on a not-quite-circle isn't much easier than just estimating where to fold while looking at a flat piece of paper. –  David Richerby Apr 5 at 12:10
I think this is a description of how everyone already folds paper, rotated 90 degrees. –  Phil Apr 5 at 17:40

This solution works only with a sheet of paper having aspect ratio of sqrt(2) (as A4 has).
Only two extra folds required.

-
• Fold bottom to middle, do not crease!
• Fold top to bottom,
• Push bottom into middle,
• Flatten paper gently,
• Crease bottom,
• Crease top.
-

A non-mathematical but quick way (which I used when I had to mail hundreds of letters years ago): Bend the paper in the Z-shape as you would want it to fold but without creasing yet, gently push the edges (top of paper and 2/3rds down on the left, 1/3rd down and bottom on the right) together so the edges align automatically, while at the same time making the pile flatter, and finally press the edges down to create the folding. You can do that with 10 sheets simultaneously and still get reasonable results.

-

Does not produce an exact solution but usually does better than guessing the first fold.

1. Roll the paper into a tube.
2. Allow an overlap to develop that looks right.
3. Carefully flatten the tube adjusting the overlap as necessary.
-

I love the geometrical solutions! That said, there is a trick for more accurate estimation. The trick is not to try to judge 1/3 of the whole sheet but to try to judge 1/2 of the remaining sheet while you are doing the first fold. In other words, as you fold the bottom 1/3 of the sheet upwards, the edge that you are moving upwards approaches the exact middle of the remaining 2/3 of the sheet. For me, this is easy enough to judge, and results in folds accurate to within a couple of millimeters.

-
That's the solution I described in my comment above. It isn't "mathematical", but it's good enough to solve the problem which motivated the question. –  keshlam Apr 7 at 2:06